737 is fairly old-fashioned from what I read about the A320:
Lateral:
HDG SEL
LNAV
VOR/LOC
LNAV follows the route in the FMC. The other two are "what it says on the tin"
Vertical:
Level Change (climbs/descends to MCP selected altitude using max N1 or idle thrust)
Vertical Speed (maintains selected VS using up to max N1 for phase of flight, speed secondary)
VNAV - several whole threads on this on its own. Follows defined vertical paths, or descends at a speed. It does its best to maintain the descent path calculated by the FMC. It varies from pretty good to utterly rubbish, and does things in what can sometimes be a rather counterintuitive way.
APP mode - follows the glideslope on the ILS
A/T Modes:
N1 (LVL CHG or VNAV SPD climb)
MCP SPD (in V/S or APP)
FMC SPD (via VNAV)
I understand that Level Change roughly equates to Open Descent in the busy, and VNAV to managed.
The Autothrottle is pretty "dumb", and requires a bit of assistance from time to time. VNAV is the mode most likely to get you wondering "what's it doing now?". VOR/LOC also doesn't track the localiser in the way a human pilot would.
If you really want the nuts and bolts, Smartcockpit.com has some good stuff on the 737 autoflight, taken from the FCOM.